First slide

Ellipse

Question

If $3 x + 4 y = 12 \sqrt{2}$ is a tangent to the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{9} = 1$ for some $a \in R$ , then the distance between the foci the of ellipse is

Moderate

A

$2 \sqrt{5}$
B

$2 \sqrt{7}$
C

4
D

$2 \sqrt{2}$

Solution

$3 x + 4 y = 12 \sqrt{2}$ $\Rightarrow 4 y = - 3 x + 12 \sqrt{2}$ $\Rightarrow y = - \frac{3}{4} x + 3 \sqrt{2}$
Condition of tangency $c^{2} = a^{2} m^{2} + b^{2}$
$\Rightarrow 18 = a^{2} . \frac{9}{16} + 9$
$\Rightarrow a^{2} . \frac{9}{16} = 9$
$\Rightarrow a^{2} = 16$
$\Rightarrow a = 4$
$\Rightarrow e = \sqrt{1 - \frac{b^{2}}{a^{2}}} = \sqrt{1 - \frac{9}{16}} = \frac{\sqrt{7}}{4}$
$∴ a e = \frac{\sqrt{7}}{4} .4 = \sqrt{7}$
$\Rightarrow$ foci are $(\pm \sqrt{7}, 0)$
$∴$ distance between foci $= 2 \sqrt{7}$

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